Suppose $A_{1}, A_{2}, . . . . . A_{30}$ are thirty sets each having 5 elements and $B_{1}, B_{2}, . . . . . B_{n}$ are n sets each with 3 elements , let $⋃_{r = 1}^{30}$ = $⋃_{f = 1}^{n} B_{j} = S$ and each of S belongs to exactly 10 of the $A_{i}^{^{'} s}$ and exactly 9 of the $B_{i}^{^{'} s}$ , then n is equal to

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Solution

The correct option is C
45

If elements are not repeated, then number of elements in $A_{1} \cup A_{2} \cup A \cup, . . . . . . \cup A_{30} \times 5$

But each element is used 10 times, so

$S = \frac{30 \times 5}{10} = 15$

If elements in $B_{1}, B_{2}, B_{n}$ are not repeated, then total number of elements in $3_{n}$ but each element is repeated 9 times, so

$S = \frac{3 n}{9} \Rightarrow 15 = \frac{3 n}{9}$

n= 45

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Similar questions

Q. Suppose

A_{1}, A_{2}, . . ., A_{30}

are thirty sets each having 5 elements and

B_{1}, B_{2}, . . ., B_{n}

are n sets each with 3 elements. Let

\cup_{i = 1}^{30} A_{i} = \cup_{j = 1}^{n} B_{j} = S

and each element of S belong to exactly 10 of the

A_{i}' s

and exactly 9 of the

B_{j}' s

, then n is equal to

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Suppose A1,A2,.....A30 are thirty sets each having 5 elements and B1,B2,.....Bn are n sets each with 3 elements , let ⋃30r=1 = ⋃nf=1Bj=S and each of S belongs to exactly 10 of the A′si and exactly 9 of the B′si, then n is equal to